An Analysis of the Influence of Surface Roughness and Clearance on the Dynamic Behavior of Deep Groove Ball Bearings Using Artificial Neural Networks

The deep groove ball bearing is one of the most important components of the rotary motion system and is the research subject in this paper. After factory assembly, new ball bearings need to pass quality control. The conventional approach relies on measuring the vibration amplitudes for each unit and sorting them into classes according to the vibration level. In this paper, based on experimental research, models are created to predict the vibration class and analyze the dynamic behavior of new ball bearings. The models are based on artificial neural networks. A feedforward multilayer perceptron (MLP) was applied, and a backpropagation learning algorithm was used. A specific method of training groups of artificial neural networks was applied, where each network provided an answer to the input within the group, and the final answer was the mean value of the answers of all networks in the group. The models achieved a prediction accuracy of over 90%. The main aim of the research was to construct models that are able to predict the vibration class of a new ball bearing based on the geometric parameters of the bearing rings. The models are also applied to analyze the influence of surface roughness of the raceways and the internal radial clearance on bearing vibrations.


Introduction
Deep groove ball bearings are one of the most important components of rotary motion systems, and the vibrations generated by the bearing have a significant influence on the dynamic behavior of the system. When designing complex rotary motion systems, whose integral parts are deep groove ball bearings, it is very important to know the dependence of vibrations and the many factors that affect them. The vibration measurement method is prescribed by the ISO 15242-2 standard. New ball bearings need to pass quality control after assembly in the factory. The conventional approach involves measuring the vibration amplitudes for each unit and sorting them into classes according to the vibration level. Mathematical models are used to describe the dynamic and static behavior of bearings [1][2][3][4]. One of the first studies of the influence of the surface roughness of raceways on bearing vibrations was conducted by Kanai et al. [5]. They developed a method for determining the surface roughness of a bearing's raceway based on an analysis of the bearing's vibration signal. Yunlong et al. [6] developed a dynamic model to determine the dynamic behavior and movement of the rolling elements of the bearing based on the surface roughness at the point contact of the ball and the raceway. Zmarzly [7] investigated the effect of surface roughness and raceway waviness on bearing vibrations. Using linear regression, the author approximated the dependence of bearing vibration amplitudes on the surface roughness and raceway waviness.
Artificial neural networks have been used for a long time for the vibration analysis of deep groove ball bearings. Their basic application is to determine the cause of bearing faults. Today, there are solutions that are practically applied for bearing fault diagnosis. A review on the application of convolutional neural networks in bearing fault diagnosis is provided in [8], and an overview of applied artificial intelligence techniques for bearing analysis and fault diagnosis was presented by Liu et al. [9]. The application of artificial neural networks for predicting vibration amplitudes of deep groove ball bearings based on technological parameters and lubrication is presented in [10]. Artificial neural networks have been successfully applied to predict the size of the internal radial clearance in deep groove ball bearings, as presented in reference [11]. Different examples of the application of artificial intelligence to detect fault in bearings are presented in [12][13][14][15][16][17]. The authors in [18][19][20][21][22] use artificial neural networks to predict the remaining useful life of the bearing (RUL).
In [23], the authors apply artificial deep learning neural networks to predict the surface roughness in metal processing. The signals were transformed using FFT (fast Fourier transformation) into a form suitable for training artificial neural networks. Lin et al. [24] analyzed metal processing parameters using artificial neural network models to predict the surface roughness of the processed surface.
Predictive maintenance is an important strategy in rotating machinery to ensure high reliability of the system. Today, there are a number of different methods that help to improve the reliability of rotary systems. The most used is analysis of measured vibration signals with different artificial intelligence tools. In [25], the authors reviewed methods used in prediction of the maintenance requirements of gearboxes, bearings and generators of wind turbines. Lalik and Watorek [26] presented a neural control algorithm for defect detection of wind turbine rotary systems. They used artificial intelligence to determine the technical condition and predict possible failure in the system.
The main aim of this research was to construct models able to predict the vibration class of a new ball bearing based on the geometric parameters of the bearing rings. The models were also applied to analyze the influence of surface roughness of the raceways and internal radial clearance on bearing vibrations. Based on experimentally obtained data, groups of artificial neural networks were trained and were used to predict the vibration amplitude of deep groove ball bearings depending on the geometric characteristics of the raceway and internal radial clearance. The axial load was constant for this experimental research and was not considered as an influential factor. In the available literature, no such approach to the investigation of the dynamic behavior of ball bearings was found. To date, this is the first study to utilize artificial neural networks to predict the vibration class of new ball bearings based on the geometric parameters of the bearing rings and to investigate the dependence of surface roughness and vibrations, as well as clearances and vibrations in the bearing.

Materials and Methods
The test sample consisted of 30 deep groove ball bearings marked 6006. The experiment involved measuring the geometric deviations and surface roughness of the raceway of the inner and outer rings, the internal radial clearance, as well as the vibration amplitudes of the ball bearings. Geometric deviations were measured using the Taylor-Hobson Form Talysurf 112/1849 device as shown in Figure 1. The statistical features of the geometric deviations and internal radial clearance are presented in Table 1. These include the internal radial clearance (G r ), raceway radius ratio (R i /R e ), surface roughness of the raceway of the outer ring (R ae ), raceway waviness of the outer ring (W te max ), deviation from the circularity of the raceway of the outer ring (W te ), surface roughness of the raceway of the inner ring (R ai ), raceway waviness of the inner ring (W ti max ), and deviation from the circularity of the raceway of the inner ring (W ti ). 21 recommends. The basic element in the vibration measurement chain is the el namics pickup (Figure 3) that generates a voltage at its output, and whose amplit frequency is proportional to the velocity of the vibrations generated by the ball The electrodynamic pickup is connected to a computer by a USB cable, which con measurement process and stores the measured signals. The measurement takes the moment the bearing reaches a constant rotation velocity. Since the amplitud signal obtained by the electrodynamic pickup used is small, the signal is amplifi amplifier for digital processing and display.    The geometric deviation parameters of the raceways for the inner and outer rings are independent since each ring is made independently of the others. When selecting the inner and outer rings for mounting the bearing, it is possible that the geometrical parameters are completely different. To eliminate the influence of different parameters, equivalent parameters were introduced, and their value was calculated using the following expressions: The equivalent surface roughness of the raceway (R a ekv ): The equivalent amplitude of raceway waviness (W t max ekv ): The equivalent amplitude of deviation from the circularity of raceway (W t ekv ): (3) Table 2 shows the main statistical characteristics of equivalent bearing parameters. The measurement of vibration velocity amplitudes was carried out using a device for measuring and analyzing bearing vibrations. Figure 2 shows the experimental device consisting of a test table on which there is a spindle mounted with hydrodynamic bearings, a pneumatic cylinder for axial loading of the bearing, a drive motor and a control cabinet (control unit). The bearing is placed over the inner ring on the measuring mandrel, which is connected to the spindle by means of a cone and a threaded connection. During the measurement, the spindle rotates at a constant number of revolutions (n = 1800 RPM) with a permissible deviation of +1% and −2%, according to ISO 15242-1. The outer ring is stationary and loaded with an axial force via a pneumatic cylinder, as standard ISO 15242-21 recommends. The basic element in the vibration measurement chain is the electrodynamics pickup ( Figure 3) that generates a voltage at its output, and whose amplitude and frequency is proportional to the velocity of the vibrations generated by the ball bearing. The electrodynamic pickup is connected to a computer by a USB cable, which controls the measurement process and stores the measured signals. The measurement takes 5 s from the moment the bearing reaches a constant rotation velocity. Since the amplitude of the signal obtained by the electrodynamic pickup used is small, the signal is amplified by an amplifier for digital processing and display.
Materials 2023, 16, x FOR PEER REVIEW is connected to the spindle by means of a cone and a threaded connection. Du measurement, the spindle rotates at a constant number of revolutions (n = 1800 RP a permissible deviation of +1% and −2%, according to ISO 15242-1. The outer rin tionary and loaded with an axial force via a pneumatic cylinder, as standard ISO 21 recommends. The basic element in the vibration measurement chain is the ele namics pickup (Figure 3) that generates a voltage at its output, and whose amplit frequency is proportional to the velocity of the vibrations generated by the ball The electrodynamic pickup is connected to a computer by a USB cable, which con measurement process and stores the measured signals. The measurement takes 5 the moment the bearing reaches a constant rotation velocity. Since the amplitud signal obtained by the electrodynamic pickup used is small, the signal is amplifie amplifier for digital processing and display.      Figure 4. The elements responsible for analogu processing are the amplifier and the bandwidth filter. The amplifier has the task o fying the signal level from the electrodynamic velocity sensor to a level suitable fo processing and display. An amplifier with a gain of 1500 was used, providing s signal amplitude for digital processing. The frequency band of the signal that is of for vibration testing of ball bearings is from 50 Hz to 10 kHz. The filter functions the spectrum of the signal received from the amplifier to the mentioned band. signed filter introduces a relatively small attenuation of wave oscillations in one frequency bands and a relatively large attenuation for oscillations of other freq (below 50 Hz and above 10 kHz) according to the ISO 15242-1 standard. The ban of the filter is also defined based on the mentioned standard. Signal digitization formed using the NI DAQ USB-6009 measurement acquisition system. The samp quency is 48 kHz, while the resolution of the internal A/D converter is 13 bits. In t the quality preparation of the signal obtained using the electrodynamic velocit and its digitization for further computer processing is ensured.
The signal recorded in the time domain is transformed into the frequency do further analysis using fast Fourier transformation. The standard requires the an vibrations in three characteristic frequency bands, namely, low frequencies (50 ÷ medium frequencies (300 ÷ 1800 Hz), and high frequencies (1800 ÷ 10000 Hz).

Analysis of Measured Data
The measurements of the geometric characteristics of the bearing raceway amplitude of the vibration velocity were analyzed to find mutual dependencies. T ysis was performed by calculating the coefficients of linear correlation between t acteristic parameters of the bearing and the amplitude of the vibration velocity in quency bands. In this analysis, the input parameters were the geometric characte the bearing raceway, and the output velocities of vibrations in characteristic ar analyzed cases were those when the geometric characteristics of the bearing racew A block diagram illustrating the measurement and control system for the vibration testing of ball bearings is shown in Figure 4. The elements responsible for analogue signal processing are the amplifier and the bandwidth filter. The amplifier has the task of amplifying the signal level from the electrodynamic velocity sensor to a level suitable for digital processing and display. An amplifier with a gain of 1500 was used, providing sufficient signal amplitude for digital processing. The frequency band of the signal that is of interest for vibration testing of ball bearings is from 50 Hz to 10 kHz. The filter functions to limit the spectrum of the signal received from the amplifier to the mentioned band. The designed filter introduces a relatively small attenuation of wave oscillations in one or more frequency bands and a relatively large attenuation for oscillations of other frequencies (below 50 Hz and above 10 kHz) according to the ISO 15242-1 standard. The bandwidth of the filter is also defined based on the mentioned standard. Signal digitization is performed using the NI DAQ USB-6009 measurement acquisition system. The sampling frequency is 48 kHz, while the resolution of the internal A/D converter is 13 bits. In this way, the quality preparation of the signal obtained using the electrodynamic velocity sensor and its digitization for further computer processing is ensured.  A block diagram illustrating the measurement and control system for the vibration testing of ball bearings is shown in Figure 4. The elements responsible for analogue signal processing are the amplifier and the bandwidth filter. The amplifier has the task of amplifying the signal level from the electrodynamic velocity sensor to a level suitable for digital processing and display. An amplifier with a gain of 1500 was used, providing sufficient signal amplitude for digital processing. The frequency band of the signal that is of interest for vibration testing of ball bearings is from 50 Hz to 10 kHz. The filter functions to limit the spectrum of the signal received from the amplifier to the mentioned band. The designed filter introduces a relatively small attenuation of wave oscillations in one or more frequency bands and a relatively large attenuation for oscillations of other frequencies (below 50 Hz and above 10 kHz) according to the ISO 15242-1 standard. The bandwidth of the filter is also defined based on the mentioned standard. Signal digitization is performed using the NI DAQ USB-6009 measurement acquisition system. The sampling frequency is 48 kHz, while the resolution of the internal A/D converter is 13 bits. In this way, the quality preparation of the signal obtained using the electrodynamic velocity sensor and its digitization for further computer processing is ensured.
The signal recorded in the time domain is transformed into the frequency domain for further analysis using fast Fourier transformation. The standard requires the analysis of vibrations in three characteristic frequency bands, namely, low frequencies (50 ÷ 300 Hz), medium frequencies (300 ÷ 1800 Hz), and high frequencies (1800 ÷ 10000 Hz).

Analysis of Measured Data
The measurements of the geometric characteristics of the bearing raceway and the amplitude of the vibration velocity were analyzed to find mutual dependencies. The analysis was performed by calculating the coefficients of linear correlation between the characteristic parameters of the bearing and the amplitude of the vibration velocity in the frequency bands. In this analysis, the input parameters were the geometric characteristics of the bearing raceway, and the output velocities of vibrations in characteristic areas. The analyzed cases were those when the geometric characteristics of the bearing raceway were observed for each ring separately and when equivalent parameters were used. Figure 5 shows the values of the linear correlation coefficient of the characteristic bearing parameters where the technological parameters are observed for the outer and inner ring separately according to the characteristic frequency bands. The linear correlation coefficients of the bearing parameters for the low-frequency band are shown in Figure 5a. The highest The signal recorded in the time domain is transformed into the frequency domain for further analysis using fast Fourier transformation. The standard requires the analysis of vibrations in three characteristic frequency bands, namely, low frequencies (50 ÷ 300 Hz), medium frequencies (300 ÷ 1800 Hz), and high frequencies (1800 ÷ 10,000 Hz).

Analysis of Measured Data
The measurements of the geometric characteristics of the bearing raceway and the amplitude of the vibration velocity were analyzed to find mutual dependencies. The analysis was performed by calculating the coefficients of linear correlation between the characteristic parameters of the bearing and the amplitude of the vibration velocity in the frequency bands. In this analysis, the input parameters were the geometric characteristics of the bearing raceway, and the output velocities of vibrations in characteristic areas. The analyzed cases were those when the geometric characteristics of the bearing raceway were observed for each ring separately and when equivalent parameters were used. Figure 5 shows the values of the linear correlation coefficient of the characteristic bearing parameters where the technological parameters are observed for the outer and inner ring separately according to the characteristic frequency bands. The linear correlation coefficients of the bearing parameters for the low-frequency band are shown in Figure 5a. The highest correlation coefficient was found for the deviation from circularity of the outer ring, followed by waviness and surface roughness of the outer ring. The next most important influencing parameter was the surface roughness of the inner ring, followed by the ratio of the radius of the raceway, the radial clearance, the deviation from the circularity and the waviness of the inner ring. rials 2023, 16, x FOR PEER REVIEW 6 of correlation coefficient was found for the deviation from circularity of the outer ring, f lowed by waviness and surface roughness of the outer ring. The next most important fluencing parameter was the surface roughness of the inner ring, followed by the ratio the radius of the raceway, the radial clearance, the deviation from the circularity and t waviness of the inner ring.  Figure 5b shows the linear correlation coefficients of the bearing parameters for t medium-frequency band. The surface roughness of the outer ring has the highest infl ence, followed by the deviation from the circularity of the outer ring, the ratio of the radi of the raceway, the waviness of the outer ring, the radial clearance, and further paramete of the inner ring of the bearing. The correlation coefficients are lower compared with t correlation coefficients in the low-frequency band.
The linear correlation coefficients of the bearing parameters for the high-frequen band are shown in Figure 5c. The values of the coefficients are significantly lower co pared with the low and medium frequency bands. The surface roughness of the inner rin the deviation from the circularity of the outer ring and the radial clearance have the hig est influence. Parameters with low influence are the surface roughness of the outer rin the ratio of the radius of the raceway, the waviness of both rings and the deviation fro the circularity of the inner ring. Figure 6 shows the linear correlation coefficients of the characteristic bearing para eters and equivalent technological parameters in the three frequency bands. Figure shows the linear correlation coefficients of the bearing parameters for the low-frequen band. The highest correlation coefficient was observed for the equivalent surface roug ness, followed by the ratio of the radius of the raceways. Equivalent waviness and equ  Figure 5b shows the linear correlation coefficients of the bearing parameters for the medium-frequency band. The surface roughness of the outer ring has the highest influence, followed by the deviation from the circularity of the outer ring, the ratio of the radius of the raceway, the waviness of the outer ring, the radial clearance, and further parameters of the inner ring of the bearing. The correlation coefficients are lower compared with the correlation coefficients in the low-frequency band.
The linear correlation coefficients of the bearing parameters for the high-frequency band are shown in Figure 5c. The values of the coefficients are significantly lower compared with the low and medium frequency bands. The surface roughness of the inner ring, the deviation from the circularity of the outer ring and the radial clearance have the highest influence. Parameters with low influence are the surface roughness of the outer ring, the ratio of the radius of the raceway, the waviness of both rings and the deviation from the circularity of the inner ring. Figure 6 shows the linear correlation coefficients of the characteristic bearing parameters and equivalent technological parameters in the three frequency bands. Figure 6a shows the linear correlation coefficients of the bearing parameters for the low-frequency band. The highest correlation coefficient was observed for the equivalent surface roughness, followed by the ratio of the radius of the raceways. Equivalent waviness and equivalent deviation  Figure 6b shows the linear correlation coefficients of the bearing parameters for the medium-frequency band. The equivalent surface roughness has the highest influence, followed by the ratio of the radius of the raceways and the radial clearance. The less influential parameters were equivalent waviness and deviation from circularity.
The linear correlation coefficients of the bearing parameters for the high-frequency band are shown in Figure 6c. The values of the coefficients were also significantly lower compared with those for the low and medium-frequency bands. The equivalent surface roughness had the highest influence, which affects the reduction of the vibration velocity amplitudes. Radial clearance was the next most important influencing parameter. The parameters with lower influence were the equivalent deviation from circularity, the ratio of the radius of the raceways and the equivalent waviness.

Application of the Neural Network
Artificial neural networks are used in this paper to investigate the influence of the surface roughness of the raceways and the clearance in the bearing on the vibration velocity amplitudes. The paper applies a multilayer perceptron (MLP), where a feedforward signal and a learning algorithm with error backpropagation are used. The neural network has one hidden layer with d inputs, k neurons in the hidden layer, and one output, and the total ratio of inputs to outputs in this case is a function of into a scalar output using the following equation [27]:  Figure 6b shows the linear correlation coefficients of the bearing parameters for the medium-frequency band. The equivalent surface roughness has the highest influence, followed by the ratio of the radius of the raceways and the radial clearance. The less influential parameters were equivalent waviness and deviation from circularity.
The linear correlation coefficients of the bearing parameters for the high-frequency band are shown in Figure 6c. The values of the coefficients were also significantly lower compared with those for the low and medium-frequency bands. The equivalent surface roughness had the highest influence, which affects the reduction of the vibration velocity amplitudes. Radial clearance was the next most important influencing parameter. The parameters with lower influence were the equivalent deviation from circularity, the ratio of the radius of the raceways and the equivalent waviness.

Application of the Neural Network
Artificial neural networks are used in this paper to investigate the influence of the surface roughness of the raceways and the clearance in the bearing on the vibration velocity amplitudes. The paper applies a multilayer perceptron (MLP), where a feedforward signal and a learning algorithm with error backpropagation are used. The neural network has one hidden layer with d inputs, k neurons in the hidden layer, and one output, and the total ratio of inputs to outputs in this case is a function of f : R d → R which maps the input vector x ∈ R d into a scalar output using the following equation [27]: Vectors w l ∈ R d contain weighting coefficients between the inputs and l-th hidden node, and v l ∈ R d is the weight coefficient of the l-th hidden node and the output. Finally, φ : R → R denotes the activation function applied to each hidden node.

Organization of Data Set
The results of the experimental research are divided into groups of I/O parameters for the application of the artificial neural network. The input parameters are the geometric characteristics of the bearing and external load, and the output results are the vibration measurements of the bearing as shown in Figure 7. Figure 8 shows the I/O parameters when the technological parameters of the bearing are considered as equivalent.

Organization of Data Set
The results of the experimental research are divided into groups of I/O parameters for the application of the artificial neural network. The input parameters are the geometric characteristics of the bearing and external load, and the output results are the vibration measurements of the bearing as shown in Figure 7. Figure 8 shows the I/O parameters when the technological parameters of the bearing are considered as equivalent.

Data Pre-Processing and Defining Datasets for Training, Validation and Testing
Prior to starting the work with neural networks, experimental data are often preprocessed which consists of linear scaling of the input data. Scaling the input parameters improves the training conditions of the networks in terms of achieving the optimal set of weight coefficients in the network. Linear scaling of the output parameters can establish

Organization of Data Set
The results of the experimental research are divided into groups of I/O parameters for the application of the artificial neural network. The input parameters are the geometric characteristics of the bearing and external load, and the output results are the vibration measurements of the bearing as shown in Figure 7. Figure 8 shows the I/O parameters when the technological parameters of the bearing are considered as equivalent.

Data Pre-Processing and Defining Datasets for Training, Validation and Testing
Prior to starting the work with neural networks, experimental data are often preprocessed which consists of linear scaling of the input data. Scaling the input parameters improves the training conditions of the networks in terms of achieving the optimal set of weight coefficients in the network. Linear scaling of the output parameters can establish

Data Pre-Processing and Defining Datasets for Training, Validation and Testing
Prior to starting the work with neural networks, experimental data are often preprocessed which consists of linear scaling of the input data. Scaling the input parameters improves the training conditions of the networks in terms of achieving the optimal set of weight coefficients in the network. Linear scaling of the output parameters can establish a balance between outputs whose intensities are very different. Linear scaling of the output parameters was performed in the interval from 0.5 to 0.8 according to Equation (5). This scaled data were further divided into sets and used to train artificial neural networks.
The data were then divided into the training set, validation set and test set. In general, the most common data distribution ratio is 70% for the training set, 15% for the validation set, and 15% for the test set. These relations can be changed to obtain a better-quality model. In this paper, for the purpose of analyzing the dynamic behavior of bearings, sets were formed according to the number of bearings in a certain set. The total number of bearings was 30 with 2 sets allocated to post-training testing and network performance evaluation, as shown in Figure 9. The remaining 28 bearings were divided into 20 for training, 4 for validation and 4 for testing. a balance between outputs whose intensities are very different. Linear scaling of the output parameters was performed in the interval from 0.5 to 0.8 according to Equation (5). This scaled data were further divided into sets and used to train artificial neural networks.
The data were then divided into the training set, validation set and test set. In general, the most common data distribution ratio is 70% for the training set, 15% for the validation set, and 15% for the test set. These relations can be changed to obtain a better-quality model. In this paper, for the purpose of analyzing the dynamic behavior of bearings, sets were formed according to the number of bearings in a certain set. The total number of bearings was 30 with 2 sets allocated to post-training testing and network performance evaluation, as shown in Figure 9. The remaining 28 bearings were divided into 20 for training, 4 for validation and 4 for testing. The evaluation of the model predictions is based on Pearson's correlation coefficient (R), the determination coefficient (R 2 ) and the average prediction error expressed as a percentage. The evaluation was calculated for each band separately. The overall evaluation was calculated as the cumulative value of the Pearson's correlation coefficient and the separate coefficients of determination, and the prediction fault was calculated as the mean value for all three bands. If the model is absolutely correct, then the values of the coefficients will be 1 in each band, that is, their cumulative value will be 3, and the mean value of the prediction fault will be zero. The model whose cumulative values of the coefficients are closest to the maximum, and at the same time, has a minimum mean value of the prediction fault, is considered the best. When selecting the model to be used for the analysis of the dynamic behavior of the bearing, the priority in the evaluation is assigned to the networks that have the smallest average prediction fault, and as a secondary evaluation, the cumulative values of the coefficients are used.

Analyzed Models of Artificial Neural Networks
There are various architectures of neural networks from those with a single hidden layer with several neurons to networks with multiple hidden layers. The training was conducted with three different training algorithms: Levenberg-Marquardt (LM), Bayesian Regularization (BR), and Scaled Conjugate Gradient (SCG). Table 3 shows the analyzed artificial neural network models. The technological parameters of the bearing raceways were analyzed separately using equivalent values. The evaluation of the model predictions is based on Pearson's correlation coefficient (R), the determination coefficient (R 2 ) and the average prediction error expressed as a percentage. The evaluation was calculated for each band separately. The overall evaluation was calculated as the cumulative value of the Pearson's correlation coefficient and the separate coefficients of determination, and the prediction fault was calculated as the mean value for all three bands. If the model is absolutely correct, then the values of the coefficients will be 1 in each band, that is, their cumulative value will be 3, and the mean value of the prediction fault will be zero. The model whose cumulative values of the coefficients are closest to the maximum, and at the same time, has a minimum mean value of the prediction fault, is considered the best. When selecting the model to be used for the analysis of the dynamic behavior of the bearing, the priority in the evaluation is assigned to the networks that have the smallest average prediction fault, and as a secondary evaluation, the cumulative values of the coefficients are used.

Analyzed Models of Artificial Neural Networks
There are various architectures of neural networks from those with a single hidden layer with several neurons to networks with multiple hidden layers. The training was conducted with three different training algorithms: Levenberg-Marquardt (LM), Bayesian Regularization (BR), and Scaled Conjugate Gradient (SCG). Table 3 shows the analyzed artificial neural network models. The technological parameters of the bearing raceways were analyzed separately using equivalent values.

Method of Training Artificial Neural Networks
The training process of the artificial neural networks was carried out for each training configuration, which includes the training algorithm, the number of hidden layers, and the number of neurons in the hidden layer. A special training algorithm was created to enable automation of the training process.

Description of the Training Algorithm
One way to improve the performance of neural networks is to train a set of neural networks on the same data. This method is used to improve the generalization of artificial neural networks when noise is present and when a small amount of data is available. For this purpose, an algorithm was written to enable the training process to be repeated n times for a certain configuration ( Figure 10). By executing the code, a set of neural networks is generated so that out of 28 bearings, the data for one are extracted for testing after training, and training is carried out with the remaining 27 bearings. The procedure is repeated until passing a cycle where all 28 bearings are separated. In this way, a set of 28 networks is obtained, and together they form one training cycle. The number of cycle repetitions can be set freely. In this paper, the maximum adopted number of repetitions is 10. Adopting more than 10 cycles requires more time for training networks, especially for multi-layered networks with 20 or more neurons in each layer. The training process was carried out according to the order listed in Table 3. Based on the number of repetitions, the results of training can range from one group of 28 networks to 10 groups of 28 networks. All of groups are tested and the best one is adopted.

Method of Training Artificial Neural Networks
The training process of the artificial neural networks was carried out for each training configuration, which includes the training algorithm, the number of hidden layers, and the number of neurons in the hidden layer. A special training algorithm was created to enable automation of the training process.

Description of the Training Algorithm
One way to improve the performance of neural networks is to train a set of neural networks on the same data. This method is used to improve the generalization of artificial neural networks when noise is present and when a small amount of data is available. For this purpose, an algorithm was written to enable the training process to be repeated n times for a certain configuration ( Figure 10). By executing the code, a set of neural networks is generated so that out of 28 bearings, the data for one are extracted for testing after training, and training is carried out with the remaining 27 bearings. The procedure is repeated until passing a cycle where all 28 bearings are separated. In this way, a set of 28 networks is obtained, and together they form one training cycle. The number of cycle repetitions can be set freely. In this paper, the maximum adopted number of repetitions is 10. Adopting more than 10 cycles requires more time for training networks, especially for multi-layered networks with 20 or more neurons in each layer. The training process was carried out according to the order listed in Table 3. Based on the number of repetitions, the results of training can range from one group of 28 networks to 10 groups of 28 networks. All of groups are tested and the best one is adopted. The process carried out in order to form a group of neural networks is shown schematically in Figure 11. The process carried out in order to form a group of neural networks is shown schematically in Figure 11.

Selection of Artificial Neural Network Models
One set consists of 28 networks that are tested on two previously separated bearings. The adopted result is the mean value of the predicted vibration velocity amplitude in all three bands for all 28 networks. The separate average values of the prediction fault for all three bands, and the cumulative values of the Pearson's correlation coefficient and the coefficient of determination are calculated for all 10 sets. Based on the calculated parameters, the sets are evaluated, and the set with the smallest fault and the highest values of Pearson's correlation coefficient and coefficient of determination is adopted.
After calculating the parameters for evaluating the prediction quality of the artificial neural network, the model with the best prediction was selected. The model with the lowest prediction fault had one hidden layer with 11 neurons, and the scaled conjugate gradient (SCG) training algorithm was then applied. With this model, the technological parameters of the bearing were observed separately. The average prediction fault of the selected model was 8.1% in each frequency band. The adopted model was further used for the purpose of predicting the quality class of a new ball bearing and analyzing the influence of the surface roughness of the raceway and the internal radial clearance on the bearing vibrations.
The best performing model, where technological parameters were considered together (equivalent), had 3 hidden layers with 15 neurons in each layer, used a Levenberg-Marquardt training algorithm and achieved a prediction fault of 8.9%.

Results and Discussion
The results given in this section apply to a constant axial load of 200 N. The other parameters for bearing geometrics are taken from bearings in the training process.

Selection of Artificial Neural Network Models
One set consists of 28 networks that are tested on two previously separated bearings. The adopted result is the mean value of the predicted vibration velocity amplitude in all three bands for all 28 networks. The separate average values of the prediction fault for all three bands, and the cumulative values of the Pearson's correlation coefficient and the coefficient of determination are calculated for all 10 sets. Based on the calculated parameters, the sets are evaluated, and the set with the smallest fault and the highest values of Pearson's correlation coefficient and coefficient of determination is adopted.
After calculating the parameters for evaluating the prediction quality of the artificial neural network, the model with the best prediction was selected. The model with the lowest prediction fault had one hidden layer with 11 neurons, and the scaled conjugate gradient (SCG) training algorithm was then applied. With this model, the technological parameters of the bearing were observed separately. The average prediction fault of the selected model was 8.1% in each frequency band. The adopted model was further used for the purpose of predicting the quality class of a new ball bearing and analyzing the influence of the surface roughness of the raceway and the internal radial clearance on the bearing vibrations.
The best performing model, where technological parameters were considered together (equivalent), had 3 hidden layers with 15 neurons in each layer, used a Levenberg-Marquardt training algorithm and achieved a prediction fault of 8.9%.

Results and Discussion
The results given in this section apply to a constant axial load of 200 N. The other parameters for bearing geometrics are taken from bearings in the training process. Table 4 presents the allowed vibration velocities of new ball bearing 6006 for low, medium and high-frequency bands. Class Q5 includes the best quality bearings, while Q7 corresponds to the lowest quality. It can be seen that there is large difference in vibration velocities between classes, and adopted models with prediction errors of less than 9% can fulfill the task and make an appropriate prediction.

Influence of the Surface Roughness of the Outer Ring
The analysis of the influence of the surface roughness of the outer ring was performed on one of two separate bearings, whereby the value of the surface roughness of the outer ring was adjusted at intervals from the minimum to the maximum value while the other parameters remained constant. Figure 12 shows the dependence of the RMS value of the vibration velocity for the three frequency bands on the amplitude of the surface roughness of the outer ring obtained by model 1. An increase in the amplitude of the surface roughness of the outer ring causes an increase in the RMS value of the vibration velocity in the lowfrequency band, and the same trend is observed in the medium-frequency band. In the band of high frequencies, the opposite trend occurs, and there is a slight decrease in the amplitude of the vibration velocity. The obtained dependencies correspond to the experimentally obtained values. In the analysis model applied, the technological parameters of the bearing were observed separately. Table 4 presents the allowed vibration velocities of new ball bearing 6006 f medium and high-frequency bands. Class Q5 includes the best quality bearings, w corresponds to the lowest quality. It can be seen that there is large difference in vi velocities between classes, and adopted models with prediction errors of less than fulfill the task and make an appropriate prediction.

Influence of the Surface Roughness of the Outer Ring
The analysis of the influence of the surface roughness of the outer ring was per on one of two separate bearings, whereby the value of the surface roughness of th ring was adjusted at intervals from the minimum to the maximum value while th parameters remained constant. Figure 12 shows the dependence of the RMS valu vibration velocity for the three frequency bands on the amplitude of the surface rou of the outer ring obtained by model 1. An increase in the amplitude of the surface ness of the outer ring causes an increase in the RMS value of the vibration velocit low-frequency band, and the same trend is observed in the medium-frequency b the band of high frequencies, the opposite trend occurs, and there is a slight decr the amplitude of the vibration velocity. The obtained dependencies correspond to perimentally obtained values. In the analysis model applied, the technological para of the bearing were observed separately.  Table 5 shows the RMS vibration velocities obtained by experimental measu and the data obtained by the model predictions. For each value, the quality class i mined according to Table 4. The results are presented for bearings with different roughness values of the outer ring raceway. The comparison of measured and pr class quality of the testing samples shows high model accuracy. In the low-fre band, all the predictions are correct. In the medium-and high-frequency bands th two misses, where the model predicted one class better quality because of the me RMS vibration velocity was close to the boundary value. According to the rule of d nation of bearing quality class, the worst class in the all the frequency bands m adopted. For test sample 1, this is Q6, even though it reached Q5 in the medium-fre band. For some predictions, there are significant differences between the predict  Table 5 shows the RMS vibration velocities obtained by experimental measurement and the data obtained by the model predictions. For each value, the quality class is determined according to Table 4. The results are presented for bearings with different surface roughness values of the outer ring raceway. The comparison of measured and predicted class quality of the testing samples shows high model accuracy. In the low-frequency band, all the predictions are correct. In the medium-and high-frequency bands there are two misses, where the model predicted one class better quality because of the measured RMS vibration velocity was close to the boundary value. According to the rule of determination of bearing quality class, the worst class in the all the frequency bands must be adopted. For test sample 1, this is Q6, even though it reached Q5 in the medium-frequency band. For some predictions, there are significant differences between the predicted and measured values, but in these cases, the model still reached the correct prediction of bearing quality, which is the main aim of this research.

Influence of the Surface Roughness of the Inner Ring
The analysis of the influence of the surface roughness of the inner ring was performed on one of two separate bearings, where the value of the surface roughness of the inner ring was adjusted at intervals from the minimum to the maximum value. The RMS values of the vibration velocity depending on the surface roughness of the inner ring raceway are shown in Figure 13. The increase in the amplitude of the surface roughness of the inner ring causes a slight increase in the amplitude of the vibration velocity in the bands of low and high frequencies. The amplitudes of the vibration velocity in the medium-frequency band have a slightly more significant growth with increasing surface roughness. In the analysis model applied, the technological parameters of the bearing were observed separately.

Influence of the Surface Roughness of the Inner Ring
The analysis of the influence of the surface roughness of the inner ring was per on one of two separate bearings, where the value of the surface roughness of th ring was adjusted at intervals from the minimum to the maximum value. The RM of the vibration velocity depending on the surface roughness of the inner ring race shown in Figure 13. The increase in the amplitude of the surface roughness of th ring causes a slight increase in the amplitude of the vibration velocity in the band and high frequencies. The amplitudes of the vibration velocity in the medium-fre band have a slightly more significant growth with increasing surface roughness analysis model applied, the technological parameters of the bearing were observe rately.  Table 6 shows the values of RMS vibration velocity obtained by experimenta urement and the values predicted by the model. For each value, the quality class mined according to Table 4. The results are presented for bearings with different l surface roughness of the inner ring raceway. The comparison of measured and pr class quality of the testing samples demonstrates the model's high accuracy.  Table 6 shows the values of RMS vibration velocity obtained by experimental measurement and the values predicted by the model. For each value, the quality class is determined according to Table 4. The results are presented for bearings with different levels of surface roughness of the inner ring raceway. The comparison of measured and predicted class quality of the testing samples demonstrates the model's high accuracy.

The Influence of Equivalent Surface Roughness and Radial Clearance
For the analysis of the influence of equivalent surface roughness and radial clearance in the bearing, in the applied model, the technological parameters of the bearing were considered as equivalent. The influence of equivalent surface roughness and radial clearance was analyzed on one of two separate bearings, where the value of the specified parameters was adjusted at intervals from the minimum to the maximum value while the other parameters remained constant.
The mutual influence of the radial clearance and the equivalent roughness amplitude on the RMS value of the vibration velocity amplitude is shown in Figure 14. The minimum vibration velocity in the low-and medium-frequency bands was ensured when the radial clearance was approximately 20 µm. The minimum vibration velocity in all frequency bands was achieved at an equivalent roughness amplitude of 0.1 µm.
predicted quality class based on the surface roughness of the inner ring raceway.

The Influence of Equivalent Surface Roughness and Radial Clearance
For the analysis of the influence of equivalent surface roughness and radial clea in the bearing, in the applied model, the technological parameters of the bearing considered as equivalent. The influence of equivalent surface roughness and radial ance was analyzed on one of two separate bearings, where the value of the specifi rameters was adjusted at intervals from the minimum to the maximum value wh other parameters remained constant.
The mutual influence of the radial clearance and the equivalent roughness amp on the RMS value of the vibration velocity amplitude is shown in Figure 14. The min vibration velocity in the low-and medium-frequency bands was ensured when the clearance was approximately 20 µm. The minimum vibration velocity in all freq bands was achieved at an equivalent roughness amplitude of 0.1 µm.

Conclusions
Based on the overall analysis of the results obtained using the neural networ els, the following conclusions can be drawn: • The adopted models are capable of predicting the quality class for new ball b and can reduce time required for quality control in bearing production; • The increase in roughness on the outer raceway causes a significant increase vibration level in the medium-frequency band (300-1800 Hz) and a mode crease in the low-frequency band (50-300 Hz), whereas the change in vibratio in the high-frequency band is negligibly small; • An increase in surface roughness on the raceway of the inner ring has a neg effect on the amplitude of the vibration velocity in the low-frequency ban causes a moderate increase in the medium and high band. The growth in the introduced parameter of the equivalent roughness of the raceway affects the ate growth in the amplitudes of the vibration velocity in the low-frequency b the medium-frequency band, the model predicts global minimum vibration ties at an equivalent roughness amplitude of 0.1 µm. In the high-frequency there is a slight decrease in the velocity of vibrations with an increase in the tude of the equivalent roughness;

Conclusions
Based on the overall analysis of the results obtained using the neural network models, the following conclusions can be drawn:

•
The adopted models are capable of predicting the quality class for new ball bearings and can reduce time required for quality control in bearing production; • The increase in roughness on the outer raceway causes a significant increase in the vibration level in the medium-frequency band (300-1800 Hz) and a moderate increase in the low-frequency band (50-300 Hz), whereas the change in vibration level in the high-frequency band is negligibly small; • An increase in surface roughness on the raceway of the inner ring has a negligible effect on the amplitude of the vibration velocity in the low-frequency band, and causes a moderate increase in the medium and high band. The growth in the newly introduced parameter of the equivalent roughness of the raceway affects the moderate growth in the amplitudes of the vibration velocity in the low-frequency band. In the medium-frequency band, the model predicts global minimum vibration velocities at an equivalent roughness amplitude of 0.1 µm. In the high-frequency band, there is a slight decrease in the velocity of vibrations with an increase in the amplitude of the equivalent roughness; • The neural network model predicted that the minimum vibration level is obtained in all frequency bands if the radial clearance has amplitude of around 20 µm and the equivalent roughness has an amplitude of around 0.05 µm.